Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211×C4 |
| direct product | C2×#14337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
19930 conjugacy classes of subgroups; only the 2114 normal subgroups are shown.
This group has 19930 conjugacy classes of subgroups — too many to draw. Only its 2114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(8,10)(15,16)(17,19)(18,20)(21,22);(12,14)(16,17)(20,21);(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,17,19,16)(18,20,22,21);(15,18,19,22)(16,20,17,21)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#616967, C2×#517592, C2×#186840, … |
| maximal normal subgroup of | C2×#616967, C2×#517592, C2×#186840, … |
| maximal quotient of | D3×#14337, C2×#154123, C2×#155625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 422439);